Weighted projective varieties can be constructed using the function projectiveVariety, just like multi- and embedded projective varieties.
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Here is a shortcut to construct weighted projective spaces:
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The object WeightedProjectiveVariety is a type, with ancestor classes MultiprojectiveVariety < MutableHashTable < HashTable < Thing.
The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/MultiprojectiveVarieties.m2:2475:0.